Second base

Number Theory Level 3

Let $b, B \geq 12$ .

If $B$ expressed in base $b$ is $12_b$ ,

and a number $x$ expressed in base $B$ is $x = 123_B$ ,

then what is the value of $x$ expressed in base $b$ ?

Notes and assumptions

We follow the custom of using letters to represent digits greater than 9. Thus, $\mathrm{A} = 10_{10}$ , $\mathrm{B} = 11_{10}$ , $\mathrm{C} = 12_{10}$ , and so on.

Note that the variable $B$ in this problem is not the same as the digit $\mathrm{B}$ .

It depends on the value of

b

x = 171_b

x = 16\mathrm{B}_b

1 solution

Arjen Vreugdenhil
Sep 22, 2017

Since $x = B^2 + 2B + 3$ and $B = b + 2$ , we have $x = (b+2)^2 + 2(b+2) + 3 = (b^2 + 4b + 4) + (2b + 4) + 3 = b^2 + 6b + 11.$ Because we know $b \geq 12$ , digits are available with values up to $\mathrm B = 11$ . Therefore the result is $x = 16\mathrm{B}_b,$ independent of the value of $b$ .

Second base

1 solution

0 pending reports